Problem: Simplify; express your answer in exponential form. Assume $z\neq 0, t\neq 0$. $\dfrac{{(z^{-4})^{3}}}{{(z^{-3}t^{-2})^{-2}}}$
To start, try working on the numerator and the denominator independently. In the numerator, we have ${z^{-4}}$ to the exponent ${3}$ . Now ${-4 \times 3 = -12}$ , so ${(z^{-4})^{3} = z^{-12}}$ In the denominator, we can use the distributive property of exponents. ${(z^{-3}t^{-2})^{-2} = (z^{-3})^{-2}(t^{-2})^{-2}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{(z^{-4})^{3}}}{{(z^{-3}t^{-2})^{-2}}} = \dfrac{{z^{-12}}}{{z^{6}t^{4}}}$ Break up the equation by variable and simplify. $\dfrac{{z^{-12}}}{{z^{6}t^{4}}} = \dfrac{{z^{-12}}}{{z^{6}}} \cdot \dfrac{{1}}{{t^{4}}} = z^{{-12} - {6}} \cdot t^{- {4}} = z^{-18}t^{-4}$.